Quantum nonlinear optics with polar J-aggregates in

microcavities

Felipe Herrera,∗,† Borja Peropadre,† Leonardo A. Pachon,†,‡ Semion K. Saikin,†,¶

and Alán Aspuru-Guzik∗,†

Department of Chemistry and Chemical Biology, Harvard University, Cambridge, USA 02138

E-mail: fherreraurbina@fas.harvard.edu; aspuru@chemistry.harvard.edu

∗To whom correspondence should be addressed†Department of Chemistry and Chemical Biology, Harvard University, Cambridge, USA 02138‡Grupo de Física Atómica y Molecular, Instituto de Física, Facultad de Ciencias Exactas y Naturales, Universidad

de Antioquia UdeA; Calle 70 No. 52-21, Medellín, Colombia.¶Institute of Physics, Kazan Federal University, 18 Kremlevskaya Street, Kazan, 420008, Russian Federation

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Abstract

We show that an ensemble of organic dye molecules with permanent electric dipole mo-

ments embedded in a microcavity can lead to strong optical nonlinearities at the single photon

level. The strong long-range electrostatic interaction between chromophores due to their per-

manent dipoles introduces the desired nonlinearity of the light-matter coupling in the micro-

cavity. We obtain the absorption spectra of a weak probe field under the influence of strong

exciton-photon coupling with the cavity field. Using realistic parameters, we demonstrate that

a single cavity photon can significantly modify the absorptive and dispersive response of the

medium to a probe photon at a different frequency. Finally, we show that the system is in

the regime of cavity-induced transparency with a broad transparency window for dye dimers.

We illustrate our findings using pseudoisocyanine chloride (PIC) J-aggregates in currently-

available optical microcavities.

2

J-aggregates1–5 are arrays of dye molecules with large dipole moments that exhibit strong

intermolecular electrostatic interaction, giving rise to collective effects in their coupling with elec-

tromagnetic fields. The specific set of linear and nonlinear optical properties of J-aggregates has

stimulated a resurgence of interest in them for applications in modern photonics. A large linear ab-

sorption cross section combined with a narrow line width6 at room temperatures make J-aggregates

attractive for the design of optical processing devices operating at low light levels. J-aggregates

can be readily coupled to solid state photonic7,8 and plasmonic9–11 structures extending the con-

ventional photonics to sub-diffraction length scales.5 An illustrative example of molecular-based

photon processing structures can be found in nature, where photosynthetic organisms use molecu-

lar aggregates to collect light and deliver the photon energy on the scale of tens of nanometers.12

Moderately strong laser fields are commonly used in free-space to observe coherent optical

phenomena in atomic gases and a few solid-state systems characterized by long dephasing times

exceeding milliseconds at room temperature.13–15 Solid-state semiconducting materials have much

shorter electronic coherence times on the order of hundreds of femtoseconds, which greatly in-

creases the laser intensity required to induce coherent optical phenomena in free space. For in-

stance, in order to observe electromagnetically-induced transparency (EIT) using inorganic quan-

tum dots with terahertz dephasing rates, the required control laser intensity should be on the order

of tens of MW/cm2.16 The same applies for organic materials, including J-aggregates. Such high

intensities can optically damage an organic medium.17 It is therefore necessary to replace the con-

trol lasers by the strong electric field per photon achievable in photonic structures,5 in order to

observe coherent optical response with organic matter at room temperature.

Experimental progress in the fabrication of organic optical microcavities has demonstrated the

ability to strongly couple an ensemble of organic chromophores with the confined electromagnetic

field of a cavity mode at room temperature,7,18–22 via the emergence of polariton modes in the cav-

ity transmission spectra. The strong coupling of organic ensembles with plasmonic modes has also

been demonstrated.5,9,23–26 Moreover, the regime of ultrastrong coupling with organic molecules

is now within reach, where the light-matter interaction energy reaches a significant fraction of the

3

associated transition frequency.27 These experimental advances enable the possibility of under-

standing and possibly manipulating the excited state dynamics of molecular aggregates using a

small number of photons.

In this Letter, we address the question whether collective multi-exciton states in J-aggregates

can be exploited for the coherent control of confined optical fields in photonic structures. In order

to achieve this, we extend the nonlinear exciton equation (NEE) formalism28,29 to account for the

non-perturbative coupling of the medium to a confined optical field. As an example, we consider

an ensemble of one-dimensional polar J-aggregate domains embedded in an optical microcavity as

a non-linear optical material with a substantial response at low light levels. We demonstrate that by

exploiting the strong dipole-dipole interaction between individual chromophores due to their per-

manent dipoles, plus the strong collective coupling of a molecular aggregate with the cavity field,

it is possible to perform light-by-light switching at the single-photon level. Specifically, we show

that the presence of a single photon at the cavity frequency can modify absorptive and dispersive

response of the organic medium to a weak external probe at a different frequency. The inter-

molecular electronic coupling between chromophores is responsible for establishing the required

anharmonicities in the material spectrum, and the large electric field per photon of the confined

cavity mode reduces the number of control photons required to achieve an observable switching

effect.

To describe the evolution of the medium polarization P(t), we employ a quantum Langevin for-

malism. The key features of our model are (i) the strong coupling of the molecular ensemble with

the cavity field, and (ii) the intermolecular resonant energy tranfer via transition dipoles, known as

Förster coupling Ji j, in addition to diagonal dipole-dipole interaction Ui j via permanent dipoles.

Additionally we consider chromophore relaxation due to spontaneous emission outside the con-

fined cavity mode, coupling of the chromophores to a phonon bath, and inhomogenous broadening

due to static disorder in chromophore transition energies. The evolution of an observable O in the

Heisenberg picture is given by dO/dt = −i[O,HS +HSB] (we use h = 1 throughout), where the

Hamiltonian HS describes the coherent evolution of the system degrees of freedom, and HSB the

4

interaction of the system with the environment. More specifically, HS describes the interaction of

a single planar J-aggregate containing N chromophores with the electromagnetic field of a single

cavity mode at frequency ωc as well as a probe field at frequency ωp, and can be partitioned as

HS = H1 +H2 +H3. (1)

The first term describes a single effective molecular aggregate, as defined in the Supporting Infor-

mation (SI), in the one-exciton eigenbasis as

H1 = ∑k

ωkB†kBk +∑

kpUkpB†

kB†pBkBp. (2)

The bosonic operator Bk annihilates an exciton in k-th mode, with k = 1,2, . . . ,N. We assume

the aggregate is a collection of two-level chromophores, with ground state |g〉 and excited state

|e〉 having a site dependent transition energy εi = ωe + di, where di is a small random shift from

the gas-phase transition frequency ωe that models structural or so-called static disorder.30,31 The

first term in eq. (2), is the diagonal form of the site-basis Frenkel exciton Hamiltonian H0 =

∑i εiB†i Bi +∑i j Ji jB

†i B j. In the point dipole approximation, the exchange coupling energy is Ji j =

(1−3cos2 Θi j)d2eg/r3

i j, where Θi j is the angle between the transition dipole moments of molecules

i and j (assumed parallel) and the intermolecular separation vector ri j = ri jri j. The second term in

eq. (2) describes the interaction between two exciton eigenstates due to long-range Coulomb forces

between the permanent dipoles of the chromophores. Here we assume a simplified form of the

scattering potential Ukp = ∑i j Ui j|cik|2|c jp|2, where cik is an element of the unitary transformation

Bi = ∑k cikBk. The interaction energy between sites is Ui j = (1−3cos2 Θi j)(∆d)2/r3i j, where ∆d =

de− dg is the change in permanent dipole moment upon excitation of the chromophores.32 For

homogeneous aggregates, large values of U12 can lead to the formation of biexcitons with a binding

energy proportional to U12.32 In this work we simplify the two-exciton problem by assuming that

the leading effect of the potential Ukp is to red-shift or blue-shift the two-exciton band with respect

to the non-interacting case, for attractive or repulsive interactions, respectively. For simplicity, we

5

take two-exciton eigenstates as direct products of single-exciton states.

The second term H2 = ωca†a+ωpE †E in eq. (1) is the free Hamiltonian for the cavity and

probe fields, and the third term describes light-matter interaction as

H3 = i∑k

gk(t)(E †Bk−B†kE )+ i∑

kqDk,kq(a†B†

kBkBq−B†qB†

kBka), (3)

where gk(t) =√

NA(~µk · ep)Ep(t) is proportional to the single-exciton transition dipole moment

~µk = 〈k|~µ|g〉 and Dk,kq =√

NA(~µk,kq · ec)Ec is proportional to the one-to-two exciton transition

dipole moment µk,kq = 〈k|~µ|kq〉. The organic medium typically consists of an ensemble of ag-

gregate domains.3,33 Within each domain the intermolecular interactions are much stronger than

between domains. For simplicity, we idealize the medium by assuming that each domain contains

a single one-dimensional aggregate, and all domains are identical. NA is the number of aggregates

in the medium (details in the SI). Ep(t) and ep are the electric field envelope and polarization of

the probe. Ec is the electric field per cavity photon and ec its polarization. The probe and cavity

polarizations are assumed to be collinear.

The system-bath interaction is partitioned as HSB =Hex+Hcav, where Hcav describes the decay

of the cavity mode through the mirror of a one-sided microcavity, which corresponds to a typical

experimental setup.17 The term Hex describes the radiative decay of excitons into electromagnetic

modes outside the cavity in addition to dephasing of excitons via interactions with phonons. The

specific form HSB and the relaxation tensors for system observables used in this work are given in

the SI. In Fig. 1 we illustrate the system under consideration and the spectrum including the two

lowest exciton bands.

We are interested in the polarization P(t) of the medium, induced by the weak coherent probe

field E , with frequency ωp. The medium polarization at frequency ωp is given by

P(t) = ∑k

µk

〈Bk(t)〉+ 〈B†

k(t)〉. (4)

We therefore need to solve the quantum Langevin equation for the exciton coherence 〈Bk(t →

6

∞)〉 in the steady state. The nonlinearity in the system Hamiltonian H1 couples the observable

Bk with an infinite hierarchy of equations of motion involving powers of the material operators

Bk and B†k . Since we are interested in the interaction of the medium with at most one probe

photon and one cavity photon on average, we invoke a Dynamics-Controlled Truncation scheme

(DCT)34 to truncate the hierarchy at third order, thus neglecting correlation functions involving

products of four or more excitonic variables. Given the small excitation density generated by

the weak probe, we also assume that the ground state population remains near unity at all times,

ignoring contributions from density terms such as 〈B†kBk〉 in the equations of motion. Moreover,

we assume a semiclassical approximation for the cavity field, which amounts to factorizing the

correlation functions involving producs of cavity and material variables. The dynamics of the

medium polarization is thus governed by the equations

ddt〈Bk〉 = (−iωk−Γk/2)〈Bk〉−gk〈E 〉+∑

pqDk,pq〈a†〉〈BpBq〉, (5)

ddt〈BpBq〉 = [−i(ωp +ωq +2Upq)−Γpq]〈BpBq〉−2∑

kDk,pq〈Bk〉〈a〉−

(gp〈Bq〉+gq〈Bp〉

)〈E 〉.

(6)

Assuming that the phonon and the photon baths are Markovian, the Langevin noise terms in the

equations of motion do not contribute to the evolution of the expectation values 〈Bk〉 and 〈a〉.35

For simplicity, we have also neglected the effect of Langevin noise terms in the two-point and

three-point correlation functions. We assume the cavity oscillator is weakly driven by a coherent

field with 〈ain〉 > 0 in the underdamped regime. We have ignored contributions of three-point

correlation functions of the form 〈B†pBqBk〉, representing one-to-two exciton coherences. These

coherence can be shown to remain negligible unless the ground state is depleated by the probe

field beyond the perturbative regime. The cavity field couples directly to the 〈B†pBqBk〉 in the

Langevin equations (see SI). Therefore, in the perturbative regime with respect to the probe field,

the cavity amplitude 〈a(t)〉 evolves as if the cavity was empty.

Despite the number of simplifications made in the derivation of equations (5) and (6), we note

7

that they have the same structure as the nonlinear exciton equations (NEE) that Chernyak et al.28,29

derived by taking into account the non-boson commutation of exciton operators, density terms, and

inelastic exciton-exciton scattering. Therefore, based on several previous studies of nonlinear opti-

cal spectroscopy using the NEE in the regime of perturbative light-matter interaction,29 we expect

our model to provide an accurate qualitative description of the nonlinear response of molecular

aggregates in microcavities where the coupling to the cavity mode is non-perturbative.

We are interested in the steady state response of the system to the probe field, for timescales

long compared with the exciton and cavity lifetimes. Moreover, we assume the cavity field decays

at a rate slower than the exciton coherence decay (1/Γk ∼ 102 fs), which does not require very

high-Q cavities at room temperature.17 This separation of timescales allows us to solve eqs. (5)-

(6) taking the cavity and probe amplitudes as constants. Another important assumption in our

model is the resolution of frequencies in the system. We require the detuning of the probe field

from the cavity field δ = ωc−ωp to be larger than the exciton linewidth. We also want the cavity

to interact resonantly with only a subset of transitions from the one-exciton band to the two-exciton

band, so that only the weak probe field can (perturbately) create excitations in the medium when

resonant with a one-exciton state. In order for the cavity not to generate excitations in the medium

when resonantly driven by an external input field 〈ain〉, we require δ < 0 and Ukp < 0 with |δ | ∼

max|Ukp|maxγk, where max|Ukp| characterizes the strength of the long-range interaction

between excitonic modes, and maxγk the exciton decay rate. Assuming the polarization P(t)

oscillates at two well-defined frequencies ωc and ωp, we use the ansatz 〈Bk(t)〉 = Xk(t)e−iωpt ,

〈BpBq〉=Ypq(t)e−i(ωp+ωc)t , 〈a〉= Ace−iωct and 〈E 〉= e−iωpt to separate eqs. (5)-(6) by frequency.

The steady state solution for the probe susceptibility is ε0χ(ωp)=∑k µkXk/iEp, with µk≡ (~µk ·ep).

The one-exciton coherences X = [X1,X2, . . . ,XN ]T are obtained by solving the linear system

MX = B, (7)

8

where B = [µ1,µ2, . . . ,µN ]T and

M =(O+2|Ac|2 T

). (8)

The N×N one-photon detuning matrix is diagonal with elements (O)nn = i∆n−Γn, where ∆n =

ωc−ωkn is the probe detuning from the n-th excitonic mode, and Γn ≡ Γkn is the decay rate.

The coupling between the one-exciton band and the two-exciton band is accounted for in the

two-photon detuning matrix T, which has diagonal elements (T)nn = ∑Nj 6=n D2

n j/(i∆n j−Γn j) and

off-diagonal elements (T)mn = DnmDmn/(i∆nm−Γnm). The elements Dnm ≡ Dkn,knkm are propor-

tional to the one-to-two exciton transition dipole matrix elements.36 The two-photon detuning

is ∆nm = ωp +ωc−ωkn −ωkm − 2Unm. In the absence of the cavity we have |Ac| = 0 and the

linear response is simply given by a sum of Lorentzians centered at the exciton frequencies ωk,

weighted by the corresponding transition dipole moments gk. The coupling to the cavity therefore

modifies the absorptive and dispersive response of the medium to the weak probe as described

below. We note that the aggregate absorption spectra obtained from eq. (7) satisfies the sum rule∫Im[χ(ωp)]dωp = Nπ/ε0 for all values of Ac.

In order to illustrate the developed model in eq. (7), we calculate the probe susceptibility χ(ωp)

for open one-dimensional homogeneous aggregates of size N. In Fig. 2 we show the computed

absorption probe spectra for PIC J-aggregates with N = 100 and N = 6. We use the transition

energy ωe = 2.25 eV for all sites, nearest-neighbour excitonic coupling J12 = −68.2 meV, and

dipole-dipole coupling U12 = −198 meV. These parameters were obtained by Markov et al.37

from the observation of a red-shifted induced absorption peak in the pump-probe spectra of PIC

aggregates. The relaxation tensors Γk and Γpq are dominated by phonon scattering (see SI), and

for simplicity we set Γk = Γpq ≈ 26 meV, which gives a lower bound for exciton dephasing rate at

room temperature.38 The vacuum Rabi frequency in microcavities can reach values on the order of

Ωc ∼ 100 meV.7 In Fig. 2 we set the vacuum Rabi frequency Ωc ≡√

NA〈e|~µ ·ec|g〉Ec = Γk so that

Dk,kp ∼√

NΩc for the strongest excitonic transitions exceeds the dissipation rates, as is required

9

in the strong coupling regime.

The probe absorption spectra in Fig. 2a displays a four-peak structure where the peak splitting

scales linearly with the mean cavity amplitude 〈a〉. The free-space spectrum corresponds to the

J-band. This trend can be qualitatively explained using a semiclassical model in which a classical

cavity field of frequency ωc couples strongly with two states in the one-exciton band (labelled

|k1〉 and |k2〉) and two states in the two-exciton band (|k1k2〉 and |k2k3〉). The coupling scheme is

illustrated in Fig. 1c. The transition dipole moments from the ground state |g〉 to the states |k1〉,

|k2〉 and |k3〉 have the largest values in the one-exciton band and satisfy µ1 > µ2 > µ3. The cavity

frequency is chosen to be on resonance with the |k1〉→ |k1k2〉 transition. The effective Hamiltonian

Heff that describes the couplings between the two bands in the rotating frame of the cavity field is

given by

Heff =

0 Ω1 0 0

Ω1 ω12−ωc Ω2 0

0 Ω2 ∆2 Ω3

0 0 Ω3 ω12 +∆23−ωc

, (9)

where the frequency parameters are defined in Fig. 1c. Energy is measured with respect to the

lowest exciton state |k1〉, i.e., ω1 ≡ 0. For a large homogeneous aggregate, the excitonic bands

become quasi-continuous and the splittings ∆2 ≡ ω2−ω1 and ∆23 ≡ ω23−ω12 become negligibly

small in comparison with typical linewidths. Therefore we can assume that the cavity strongly

couples almost on resonance the four levels shown in Fig. 1c. In this regime, a weak probe field

will drive transitions between the ground state |g〉 and the eigenvalues of the effective Hamiltonian

Heff in eq. (9), which are the new normal modes of the cavity-matter system. In the inset of Fig.

2a we show the eigenvalues of Heff for ωc = ω12, ∆2 = ∆23 = 0, Ω2 = 3Ω1 and Ω3 = Ω2/3 as a

function of Ωc ≡ Ω2. For Ωc 1 (in arbitrary units) only three peaks can be resolved, but as Ωc

increases the middle peak splits into a doublet, giving rise to the four-peak structure observed in the

probe spectra calculated using eq. (7), which includes N one-exciton states and ∼ N2 two-exciton

10

states.

The number of states in the one and two-exciton bands that couple strongly with the cavity

field depends on the size of the molecular aggregate. In order to illustrate this fact, we show in

Fig. 2b the probe absorption spectrum for an open 1D homogeneous aggregate of size N = 6, with

the same values of ωe, J12 and U12 and decay parameters as in Fig. 2a. The cavity field is again

resonant with the |k1〉 → |k1k2〉 transition, but now the states |k2〉 and |k3〉 are no longer quasi-

degenerate with |k1〉 because of the small array size. The cavity frequency is thus detuned from

their corresponding transitions with the states |k1k2〉 and |k2k3〉 in the two-exciton band. Since

the Rabi frequency Ω3 is proportional to the transition dipole µk2,k2k3 by construction, whenever

Ω3/∆23 1, we can set Ω3 = 0 in eq. 9 to effectively remove the state |k2k3〉 from the excited state

dynamics. Interestingly, the eigenstates of the resulting three-level system with ∆2 > 0, plotted as

a function of Ωc in the inset of Fig. 2b, show a trend in very good agreement with the microscopic

model derived in eq. (7).

We now include the effect of inhomogeneous disorder in the evaluation of the probe absorption

spectra. In order to model static energetic disorder we assume the site energies are given by εi =

ωe + di, where di is a random energy shift taken independently for each site from a Gaussian

distribution with standard deviation σ/|J| ∼ 0.1, consistent with the motional narrowing limit.30,31

The susceptibility χp obtained from eq. (7) needs to be averaged over the ensemble of disorder

realizations. In Fig. 3 we show the probe absorption spectrum A (ωp) of the same J-aggregates

used in Fig. 2a with N = 100 and N = 6 molecules, but now static disorder is introduced in the

Hamiltonian. The structure of the spectrum resembles the results in Fig. 2a, but the details of the

splittings near the origing are different because now the detunings ∆2 and ∆23 in eq. (9) are now

averaged over an ensemble of disorder realizations. The behaviour of the outer peaks persist in the

presence of disorder as a result of the strong interaction between the cavity mode and the transition

|k1〉 → |k1k2〉, close to the deterministic resonance frequency ω12 = ω2−2|U12|.

As a final example, we consider the response of a dimer of coupled polar chromophores.39

Clearly for N = 2 the bosonic approximation used in the derivation of eq. (7) is no longer valid

11

to describe the two-exciton manifold, which now consists of a single state |e1e2〉. However, since

the structure of eqs. (5) and (6) is universal, the steady-state solution X in eq. (7) remains valid

by simply redefining the elements of the two-photon detuning matrix T. The one-exciton manifold

has states |ψ+〉=√

a|e1g2〉+√

1−a|g1e2〉 and |ψ−〉=−√

1−a|e1g2〉+√

a|g1e2〉 with 0≤ a≤

1. The transition dipole moments from the one-exciton manifold to the ground and two-exciton

states are given by µ± ≡ 〈ψ±|~µ|g1g2〉 = µeg(√

a±√

1−a)= 〈ψ±|~µ|e1e2〉. Given that the two-

exciton energy ω12 = epsilon1+ε2−|U12| is red-shifted with respect to the one-exciton transition

frequencies ω±, the response of a single dimer to a probe field at frequency ωp, when the cavity is

resonant with the |ψ+〉 → |e1e2〉 transition, is given by

χ(ωp) = i[

µ21

ε0

](Γ12− i[∆p +∆c])

(i∆p−Γ1)(i[∆p +∆c]−Γ12)+2D212|Ac|2

, (10)

which corresponds to eq. (7) in the limit where M has a single non-zero element. Here ∆p =

ωp−ω+, ∆c = ωc−(ε1+ε2+U12−ω+), µ1 = µ+ and D12 ∝ µ+. Γ1 and Γ12 are the one and two-

exciton decoherence rates. Equation (10) shows that for coupled dimers the cavity acts as a control

for the propagation of the weak probe, in analogy with the phenomenology stemming from atomic

physics to describe electromagnetically-induced transparency (EIT) in a cascaded three-level sys-

tem.13 Figure 4 shows the absorptive and dispersive response of an inhomogeneously broadened

dimer of polar chromophores in panels a and b, respectively. The probe susceptibility has the

standard features of EIT: reduction of probe absorption and steep dispersion on resonance with

the lowest exciton state.13 In comparison with atomic systems, the EIT linewidth for the dimer is

broader even in the absence of static disorder because the two-exciton coherence is short-lived, i.e.,

Γ12/Γ1 ∼ 1. Inhomogeneous broadening further broadens the EIT features, and in particular in-

creases the absorption minimum under conditions of one and two-photon resonances ∆p = 0 = ∆c.

For homogeneously broadened dimers (described by eq. (10)), the absorption minimum for a res-

onant probe is plotted in Fig. 4c, showing a scaling of Amin ≈ (Γ12/2D212)A

−2c for large cavity

coupling D12Ac Γ1 ∼ Γ12. The homogeneous curve gives a lower bound for the probe absorp-

12

tion minimum on resonance. The strength of the static disorder in the site basis is given by σ as

before. We average over an ensemble of 1200 realization of the site energy shifts (d1,d2) using

an uncorrelated Gaussian joint probability distribution (JPD) of the form P(d1,d2) = P(d1)P(d2).

Increasing the disorder strength σ , increases the resonant absorption of the probe Amin for interme-

diate values of D12Ac. However, as the strength of the cavity coupling increases the homogeneous

limit is recovered. This behaviour has already been observed for EIT in Doppler-broadened atomic

gases.40

In order to gain qualitative analytic understanding of the absorption minimum Amin for inho-

mogeneously broadened dimers with σ 6= 0, we evaluate the mean susceptibility 〈χ(ωp)〉 directly

from eq. (10) by averaging over an ensemble of one- and two-exciton detunings ∆p =∆(0)p −Dp and

∆c = ∆(0)c −Dc, where the random shifts (Dp,Dc) ultimately result from the site energetic disorder

(d1,d2). We integrate over all possible shifts using 〈χ(ωp)〉=∫ ∫

dDpdDc χ(ωp,Dp,Dc)P(Dp,Dc),

where P(Dp,Dc) is the JPD for the one- and two-exciton shifts. In general, P(Dp,Dc) does not

factorize even if P(d1,d2) does, due to Förster coupling Ji j.31 In order to simplify the integra-

tion over disorder, we assume an uncorrelated JPD of the form P(Dp,Dc) = P(Dp)P(Dc), where

P(D) = π−2γ/(γ2 +D2) is the Cauchy distribution with width γ . The probe absorption under

conditions of deterministic one- and two-photon resonances ∆0p = 0 = ∆0

c thus gives

Amin =Γ12 + γp + γc

(Γ1 + γp)(Γ12 + γp + γc)+2D212A2

c, (11)

where γp and γc are the widths of P(Dp) and P(Dc), respectively. We find that Amin in eq. (11) for

a Cauchy distribution provides an upper bound for the results obtained by numerically averaging

the independent site disorder over an Gaussian distribution with the same width. However, in the

limit Ac (Γ1 + γp)/D12, eq. (11) gives Amin ≈ [(Γ12 + γp + γc)/2D212]A

−2c , which tends towards

the homogeneous limit for short-lived two-exciton coherences Γ12 (γp + γc), which is the case

considered here for dimers.

In summary, we present in this Letter a general scheme to perform nonlinear optical experi-

13

ments using polar J-aggregates at the single-photon level. The setup involves the use of organic

chromophores with a moderate to large difference between ground and excited state permanent

dipole moments ∆d, that can assemble into low-dimensional aggregate structures. We have illus-

trated our findings using pseudoisocyanine chloride (PIC) dyes, but the conclusions of this work

are general. Organic chromophores with large ∆d ∼ 1− 10 Debye continue to be under active

experimental investigation for the design of second-order nonlinear optical materials.41–43 Upon

aggregation, these polar dyes can lead to strong exciton-exciton interactions that exceed the broad-

ening of the exciton line. For attractive interactions (J-aggregation), the cavity field can be used to

strongly drive coherences between the one- and two-exciton bands without removing population

from the ground state of an aggregate. Under these conditions the absorption of a weak probe field

resonant with the cavity-free exciton absorption peak is significantly modified by the presence of

the cavity field containing a single photon (on average), which can be seen as quantum optical

switching. In order to achieve this effect it is important that the cavity-matter coupling exceeds

all the dissipation rates in the system, a regime that is experimentally accessible.7,20 We have re-

stricted our discussion to strong light-matter coupling in optical microcavities, but the strong cou-

pling regime has also been achieved for molecular aggregates in the near-field of plasmonic nanos-

tructures,9,10,25 which further opens the applicability of our proposed scheme to sub-wavelength

nonlinear quantum optics.

The ability to control molecular aggregates in optical nanostructures not only offers opportuni-

ties for the development of novel organic-based optical devices,5 but we envision new possibilities

of quantum control of excited state dynamics relevant in energy transport and chemical reactiv-

ity, and engineering of excitonic materials that are topologically robust against disorder.44 Current

experiments can achieve the regime of ultrastrong coupling with organic ensembles, where the

light-matter interaction strength can be a significant fraction of the chemical binding energy.27

In this regime, it should be possible to control the outcome of chemical reactions at the level of

thermodynamics by effectively lowering reaction barriers,45 in analogy with traditional catalytic

processes, thus directly affecting reaction kinetics. This novel strong-field single-photon quantum

14

control paradigm for molecular processes should be contrasted with traditional strong-field laser

control schemes that require very high laser intensities to modify the chemical energy landscape,46

or weak-field coherent control schemes that exploit delicate laser-induced quantum interferences

among internal vibronic states,47 but do not modify the energetics of the reaction. Quantum op-

tical control of chemical dynamics is a future research direction with promising applications in

nanoscience and technology, where traditional bulk methods for controlling chemical reactivity

have limited efficiency.

Acknowledgement

We thank Frank Spano and Thibault Peyronel for discussions. F.H. and A.A.-G. acknowledge the

support from the Center for Excitonics, an Energy Frontier Research Center funded by the U.S. De-

partment of Energy, Office of Science and Office of Basic Energy Sciences, under Award Number

de-sc0001088. F.H., S.K.S. and A.A.G. also thank the Defense Threat Reduction Agency Grant

HDTRA1-10-1-0046. S.K.S. is also grateful to the Russian Government Program of Competitive

Growth of Kazan Federal University. L.A.P. acknowledges support from the Comité para el De-

sarrollo de la Investigación -CODI– of Universidad de Antioquia, Colombia under the Estrategia

de Sostenibilidad 2014-2015, and by the Departamento Administrativo de Ciencia, Tecnología e

Innovación –COLCIENCIAS– of Colombia under the grant number 111556934912.

Supporting Information Available

The derivation of the quantum Langevin equations leading to eqs. (5)-(6), and the derivation of the

effective Hamiltonian in eq. (9) can be found in the Supporting Information (SI).

This material is available free of charge via the Internet at http://pubs.acs.org/.

15

(a)

(b)

(c)

Figure 1: Panel (a): Illustration of an optical microcavity containing an ensemble of two-dimensional J-aggregates. The cavity mode ac is driven by a weak input field ain and decaysthrough the semi-reflecting mirror at the rate γc. A weak probe field at frequency ωp > ωc cou-ples directly to the organic chromophores. Individual molecules decay into external modes witha rate γe. Dipole-dipole interactions between individual chromophores in each aggregate modifythe single-molecule response of the medium to the cavity and probe fields. Panel (b): Energyspectrum of an individual aggregate showing the one-exciton and two-exciton bands (bandwidthsnot on scale). The cavity field drives all the allowed coherences between states |ki〉 and |kik j〉and the weak probe removes population from the ground state |g〉. The transition frequency be-tween |ki〉 and |kik j〉 is shifted by the interaction energy ∼Ui j with respect to the non-interactingcase. Panel (c): Effective four-level system interacting coupled by the cavity with Rabi frequen-cies Ω2 > Ω1 > Ω3. The cavity frequency ωc is assumed to be near resonance with the transition|k1〉 → |k2〉. This model is used in eq. (9) to describe the probe absorption at frequency ωp < ωc.

16

-10 0 10Δ1/γ

-20

0

20

40

60

Im[χ]

0 0.5 1 1.5 2-1

0

1

a

b

cd

e

(a)

-5 0 5 10Δ1/γ

-5

0

5

10

15

20

Im[χ]

0 0.5 1-0.5

0

0.5

1

a

b

c

d

e

(b)

Figure 2: Probe absorption spectrum A (ωp) = −Im[χ] (in units of ε0/h) as a function of thedetuning from the lowest exciton resonance ∆1 = ωp−ω1 (in units of the decay rate Γ = 26 meV)for an ideal open linear polar J-aggregate of size N in a microcavity (no energetic disorder). Panela: N = 100. Curves are labeled according to the mean cavity amplitude Ac ≡ |〈a〉| as (a) No cavity,(b) Ac = 0.2, (c) Ac = 0.4, (d) Ac = 0.6, (e) Ac = 0.8. The inset shows the eigenvalues of thefour-level effective Hamiltonian in eq. (9) with ∆2 = 0 = ∆23 and Ω1 = Ω2/3 = 3Ω3 as a functionof Ωc = Ω2 (in arbitrary units). Panel b: N = 6. Curves are labelled by the value of Ac as (a) Nocavity, (b) Ac = 0.4, (c) Ac = 0.8, (d) Ac = 1.2, (e) Ac = 1.6. The inset shows the eigenvalues ofthe eq. (9) with ∆2 = 0.5, Ω1 = Ω2/3 and Ω3 = 0 as a function of Ωc = Ω2. In both panels thecavity frequency is resonant with the transition |k1〉 → |k1k2〉.

17

-10 -5 0 5 10

0

50

100

150

200

250

-10 -5 0 5 100

1

2

3

ab

cd

N = 6 (b)

(a)

a

ef

bcd

Figure 3: Probe absorption A (ωp) for a 1D J-aggregate as a function of the probe detuning froman arbitrarily chosen one-exciton state near the bottom of the one-exciton band. Dipolar couplingsJi j and Ui j are the same as in Fig. 2a. The cavity frequency is chosen such that the two-photondetuning vanishes when ωp−ω1 = 0. Panel a: Aggregate size is N = 100. Curves are labeledaccording to the mean cavity amplitudes: (a) No cavity, (b) Ac = |〈a〉|= 0.1, (c) Ac = 0.2, (d) Ac =0.3. Panel b: Aggregate size is N = 6 for (a) No cavity, (b) Ac = 0.4, (c) Ac = 0.8, (d) Ac = 1.2,(e) Ac = 1.6, (f) Ac = 2.0. In both panels the vacuum Rabi frequency is Ωc = Γ, where Γ = 26 meVis the exciton decay rate. Static disorder is modelled by taking each monomer energy randomlyfrom a Gaussian distribution with mean E0 = 2.25 eV and standard deviation σ = 0.125J. a widefrequency range for a low number of cavity photons. γe is the single-molecule gas-phase radiativedecay rate.

18

-6 -4 -2 0 2 4 6

0

1

2

3

4

5

-6 -4 -2 0 2 4 6

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

a

b

c

a

b

c

(a)

(c)

(b)

Figure 4: Im [χp] (panel a) and Re[χp] (panel b) for an inhomogeneously broadened dimer as afunction of the probe detuning from the maximum of free-space exciton absorption band. Dipolarcouplings Ji j and Ui j are the same as in Fig. 2. The cavity frequency is chosen such that the two-photon detuning vanishes when ωp−ω1 = 0. Curves are labeled according to the mean photonnumber Ac = |〈a〉|: (a) No cavity, (b) Ac = 0.5, (c)Ac = 1.0. The vacuum Rabi frequency isΩc = 5Γ, where Γ = 26 meV is the exciton decay rate. Panel c: Minimum absorption Amin(normalized to the free-space value) near the deterministic resonance ωp−ω1 = 0 (between theAutler-Townes doublet), as a function of the mean cavity amplitude Ac. Curves are labelled bythe width of the Gaussian distribution of static energy shifts σ (in units of the exchange dipolecoupling J). All other parameters are the same as in panels a and b.

19

Notes and References

(1) Jelley, E. E. Nature 1936, 138, 1009–1010.

(2) Scheibe, G. Angew. Chem. 1936, 49, 563.

(3) Kobayashi, T. J-aggregates; World Scientific, 1996.

(4) Agranovich, V. M. Excitations in organic solids; International Series of Monographs in

Physics; Oxford University Press, 2008.

(5) Saikin, S. K.; Eisfeld, A.; Valleau, S.; Aspuru-Guzik, A. Nanophotonics 2013, 221, 22–38.

(6) Würthner, F.; Kaise, T. E.; Saha-Möller, C. R. Angew. Chem. Int. Ed. 2011, 50, 3376–410.

(7) Lidzey, G. D. et al.. Nature 1998, 395, 53–55.

(8) Coles, D.; Somaschi, N.; Michetti, P. Nature Materials 2014, 13, 712–719.

(9) Bellessa, J.; Bonnand, C.; Plenet, J. C.; Mugnier, J. Phys. Rev. Lett. 2004, 93, 036404.

(10) Vasa, P.; Wang, W.; Pomraenke, R. Nature Photon 2013, 7, 1–5.

(11) Zengin Gülis,; Johansson Göran,; Johansson Peter,; Antosiewicz Tomasz J.,; Käll Mikael,;

Shegai Timur, Sci. Rep. 2013, 3, 3074.

(12) Scholes, G. D.; Fleming, G. R.; Olaya-Castro, A.; van Grondelle, R. Nature Chemistry 2011,

3, 763–74.

(13) Fleischhauer, M.; Imamoglu, A.; Marangos, J. P. Rev. Mod. Phys. 2005, 77, 633–673.

(14) Shore, B. Acta Physica Slovaca. Reviews and Tutorials 2008, 58, 243–486.

(15) Lee, K. C.; Sprague, M. R.; Sussman, B. J.; Nunn, J.; Langford, N. K.; Jin, X.-M.; Cham-

pion, T.; Michelberger, P.; Reim, K. F.; England, D.; Jaksch, D.; Walmsley, I. A. Science

2011, 334, 1253–1256.

20

(16) Houmark, J.; Nielsen, T. R.; Mork, J.; Jauho, A.-P. Phys. Rev. B 2009, 79, 115420.

(17) Akselrod, G. M.,; Tischler, Y. R.,; Young, E. R.,; Nocera, D. G.,; Bulovic, V., Phys. Rev. B

2010, 82, 113106.

(18) Schouwink, P.; Berlepsch, H.; Dähne, L.; Mahrt, R. Chemical Physics Letters 2001, 344,

352–356.

(19) Tischler, J. R.; Bradley, M. S.; Bulovic, V.; Song, J. H.; Nurmikko, A. Phys. Rev. Lett. 2005,

95, 036401.

(20) Kena-CohenS.,; Forrest, S. Nat Photon 2010, 4, 371–375.

(21) Kéna-Cohen, S.; Davanço, M.; Forrest, S. R. Phys. Rev. Lett. 2008, 101, 116401.

(22) Bittner, E. R.; Zaster, S.; Silva, C. Phys. Chem. Chem. Phys. 2012, 14, 3226–3233.

(23) Fofang, N. T.; Park, T.-H.; Neumann, O.; Mirin, N. A.; Nordlander, P.; Halas, N. J. Nano

Letters 2008, 8, 3481–3487.

(24) Dintinger, J.; Klein, S.; Bustos, F.; Barnes, W. L.; Ebbesen, T. W. Phys. Rev. B 2005, 71,

035424.

(25) Wurtz, G. A.; Evans, P. R.; Hendren, W.; Atkinson, R.; Dickson, W.; Pollard, R. J.; Zay-

ats, A. V.; Harrison, W.; Bower, C. Nano Letters 2007, 7, 1297–1303.

(26) Sugawara, Y.; Kelf, T. A.; Baumberg, J. J.; Abdelsalam, M. E.; Bartlett, P. N. Phys. Rev. Lett.

2006, 97, 266808.

(27) Schwartz, T.; Hutchison, J. A.; Genet, C.; Ebbesen, T. W. Phys. Rev. Lett. 2011, 106, 196405.

(28) Chernyak, V.; Zhang, W. M.; Mukamel, S. The Journal of Chemical Physics 1998, 109,

9587–9601.

(29) Mukamel, S.; Abramavicius, D. Chemical Reviews 2004, 104, 2073–2098.

21

(30) Knapp, E. Chemical Physics 1984, 85, 73–82.

(31) Knoester, J. J. Chem. Phys 1993, 99, 8466.

(32) Spano, F. C.; Agranovich, V.; Mukamel, S. The Journal of Chemical Physics 1991, 95, 1400–

1409.

(33) Kato, N.; Saito, K.; Aida, H.; Uesu, Y. Chemical Physics Letters 1999, 312, 115–120.

(34) Portolan, S.; Di Stefano, O.; Savasta, S.; Rossi, F.; Girlanda, R. Phys. Rev. B 2008, 77,

195305.

(35) Gardiner, C. W. Quantum Noise; Springer-Verlag, 1991.

(36) In the bosonic approximation for the exciton states, we evaluate these matrix elements us-

ing the eigenstates |k〉 = ∑i cik|ei〉 as 〈k|µ|kq〉 = µeg ∑i j c∗ikc jkciq + c∗ikcikc jq, where we have

defined the dipole operator in the site basis as µ = µeg ∑m |em〉〈gm|+ |gm〉〈em|.

(37) Markov, R.; Plekhanov, A.; Shelkovnikov, V.; Knoester, J. Physica Status Solidi B Basic

Research 2000, 221, 529–534.

(38) Valleau, S.; Saikin, S. K.; Yung, M.-H.; Guzik, A. A. The Journal of Chemical Physics 2012,

137, 034109.

(39) Halpin Alexei,; M., J. J.; Tempelaar Roel,; Murphy R. Scott,; Knoester Jasper,; C., J. L.;

Dwayne, M. J. Nat Chem 2014, 6, 196–201.

(40) Gea-Banacloche, J.; Li, Y.-q.; Jin, S.-z.; Xiao, M. Phys. Rev. A 1995, 51, 576–584.

(41) Verbiest, T.; Houbrechts, S.; Kauranen, M.; Clays, K.; Persoons, A. J. Mater. Chem. 1997, 7,

2175–2189.

(42) Luo, J.; Zhou, X.-H.; Jen, A. K.-Y. J. Mater. Chem. 2009, 19, 7410–7424.

(43) Li, W.; Zhou, X.; Tian, W. Q.; Sun, X. Phys. Chem. Chem. Phys. 2013, 15, 1810–1814.

22

(44) Yuen-Zhou, J.; Saikin, S.; Yao, N.; Aspuru-Guzik, A. arXiv:1406.1472

(45) Hutchison, J. A.; Schwartz, T.; Genet, C.; Devaux, E.; Ebbesen, T. W. Angewandte Chemie

International Edition 2012, 51, 1592–1596.

(46) E., C.; González-VázquezJ.,; BalerdiG.,; R., S.; R., d. N.; BañaresL., Nat Chem 2014, 6,

785–790.

(47) Shapiro, M.; Brumer, P. Quantum Control of Molecular Processes; Wiley-VCH, 2011.

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